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Interactive Causes: Revising the Markov Condition
Gerhard Schurz
Abstract: This paper suggests a revision of the theory of causal nets (TCN). In Section 1
we introduce an axiomatization of TCN based on a realistic understanding. It is shown that
the causal Markov condition entails three independent principles. In Section 2 we analyze
indeterministic decay as the major counterexample to one of these principles: screening-off
by common causes (SCC). We call (SCC)-violating common causes interactive causes. In
Section 3 we develop a revised version of TCN, called TCN*, which accounts for interac-
tive causes. It is shown that there are interactive causal models that admit of no faithful
non-interactive reconstruction.
Contact address:
Professor Gerhard Schurz
Department of Philosophy (DCLPS), Heinrich Heine University Duesseldorf
Universitaetsstrasse 1, Geb. 24.52
40225 Duesseldorf, Germany
E-mail: schurz@phil.uni-duesseldorf.de
Acknowledgement: This work was supported by the DFG (Deutsche Forschungsgemein-
schaft), research unit FOR 1063 (Causation, Laws, Dispositions, Explanation). For valua-
ble help I am indebted to Christopher Hitchcock, Paul Näger and an unknown referee.
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Interactive Causes: Revising the Markov Condition
Gerhard Schurz
1. Introduction: The Received Theory of Causal Nets
One of the best developed contemporary accounts of causality is the theory of causal nets,
for short: TCN. While most earlier accounts attempted to explicate causality by means of
definitions, the TCN account characterizes causality via axiomatic principles that explain
probabilistic relations of (in)dependence between variables by the existence of directed
causal relations between them. This realistic understanding of TCN is informally acknowl-
edged by SGS (1993, sec. 3.4-5) and Pearl (2009, preface, viii-ix), although both Pearl and
SGS presents TCN's axioms as definitions (SGS speak of "axioms" on p. 4 and 29). An
explicitly axiomatic reconstruction of TCN is given in Schurz and Gebharter (2016).
This section gives a brief exposition of the notions and axioms of TCN. A (mathemati-
cal) variable is a measurable function X: DVal(X) from a domain D of individuals to its
value space Val(X) = {x1,x2,}, which is a family of properties or a set of numbers. If X
denotes color, for example, then Val(X) = {red, green,} and X assigns a color X() to
every individual D. D may also consist of n-tuples of individuals, e.g., individuals at
certain time-points. Binary properties are represented by binary variables XF with value
space {F,F}. Terminological conventions:
X,Y,Z (possibly indexed) are variables and U,V,W are sets of variables;
lower-case letters "x" (or "xi") stand for values of X, and lower-case "u" (or "ui") for sets
of values of the respective variables in U.
A causal graph (a CG for short) is a pair (V,E), where V = {X1, X2,} is a set of vari-
ables (the "vertices") and E V a set of directed arrows XiXj (the "edges").
1
A graph
(V,E) together with a probability-distribution P is called a causal model (a CM) (V,E,P).
When we use causal graphs or models to represent reality we speak of causal structures or
systems, respectively.
P is an objective probability distribution over a suitable algebra AL over the value
space over V, i.e., P: AL(iIVal(Xi))[0,1] (with corresponding projections to subspac-
es). Further terminological conventions:
"P(x)" (or "P({x})") abbreviates "P(X()=x)", i.e., the probability that X takes value x in
the underlying domain D (the individual variable is bound by P).
Likewise, "P(S)" stands for "P(X()S)", "P(x)" for "P(X() x)", "P(x,y)" for
"P(X()=x Y()=y)", and "P(x|y)" for "P(X()=x|Y()=y)", i.e., the conditional proba-
1
The variables in V are ordered by indices X1, X2, in order to define the Cartesian
product of their value space, iIVal(Xi); these indices have no substantial meaning.
When applying the probability operator P to sets of variables we assume the variables
to be ordered according to their indices.
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bility of x given y, provided P(y) > 0.
Two variables X,Y are probabilistically dependent (DEP(X,Y)) iff some of their values
are dependent; otherwise X, Y are probabilistically independent. More generally, probabil-
istic (in)dependence between X and Y conditional on (fixed values of) a set of variables U
is defined as follows:
DEP(X,Y|U) iff x,y,u: P(x|y,u) P(x|u) and P(y,u) > 0, and
INDEP(X,Y|U) iff not DEP(X,Y|U), i.e., iff x,y,u: P(x|y,u) = P(x|u) or P(y,u) = 0.
If the variables' values are ordered according to size, the notion of positive/negative
dependence between variables can be defined as follows: X depends positively/negatively
on Y iff P(x1>x2|y1>y2) >/< P(x1>x2) (for all x1, x2, y1, y2).
A CG (V,E) is described by using the following notions:
XY: X is a direct cause of Y (Y is a direct effect of X).
XY: X is a causal ancestor
2
of Y (Y is a causal successor of X), i.e., there is a
directed path XZ1ZnY from X to Y (for n0).
XY: XY or XY, i.e., X and Y are adjacent.
X1 Xn: a path X1Xn between X1 and Xn that connects X1 and Xn; the variables
Xi (1 i n) are said to lie on this path.
If Xi lies on a path , then Xi is called (i) a common cause, (ii) an intermediate cause, or
(iii) a common effect on iff (i) X, (ii) X or X, or (iii) X, respectively,
is part of .
The set of all direct causes of a variable X is called the set of its parents, Par(X). Note
that TCN is compatible with ontological indeterminism, i.e., the values of Par(X) need not
determine the value of X. Determinism in regard to variable X is given when for all possi-
ble values par(X) Val(Par(X)) there exists an xVal(X) with P(x|par(X)) = 1.
TCN explains probabilistic dependencies via causal connections by means of two core
axioms: causal d-connection (C) and minimality (Min). (C) is frequently expressed by the
equivalent causal Markov condition (M) (SGS 2000, sec. 3.4.1). We prefer (C) over (M)
for reasons specified below. The axiom (C) states that every (conditional) probabilistic
dependence between two variables X and Y is the result of a causal "d-connection" bet-
ween X and Y:
(1) Definition: A CM (V,E,P) satisfies the condition (C) of causal d-connection iff for all
X,YV and U V{X,Y}:
If DEP(X,Y|U), then X and Y are d-connected given U in the following sense: X and Y
are connected by some path π such that:
(SIC): no intermediate cause on is in U,
(SCC): no common cause on is in U, and
(CE): every common effect on π is in U or has a causal successor in U.
Axiom: Every physically possible causal system that is [sufficiently] complete satisfies
2
The notion of "causal ancestor" does not imply that causation is transitive (but faithful
causation is transitive; see below).
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condition (C).
In (1) we distinguish between the definition of the causal d-connection condition and
the corresponding axiom, which states that this condition holds for all physically possible
systems that are either complete or at least sufficiently complete (as indicated by the square
brackets); these two notions of completeness are characterized below.
Two variables X, Y are "d-separated" given U if they are not d-connected given U. The
concepts of d-separation and d-connection were developed by Pearl (1988, 117). Axiom
(C) asserts an implication from (probabilistic) dependence to d-connection, or in con-
traposed form, from d-separation to independence (Pearl 1988, 119; 2000, th. 1.2.4). Axi-
om (C) implies the standard causal principle for unconditional dependence: If DEP(X,Y),
then X and Y are connected by a directed or common cause path (i.e., d-connected by the
empty set ).
Our formulation in (1) makes it plain that condition (C) entails three independent prin-
ciples:
(SIC) (for "screening off by intermediate causes"): Direct causes screen off indirect
causes from their effects. Thus in a [sufficiently] complete CM of the form XZY, IN-
DEP(X,Y|Z) holds. Condition (SIC) goes back to Andrei Markov (senior).
(SCC) (for "screening off by common causes"): Common causes screen off their effects
from each other. So in a [sufficiently] complete CM of the form XZY, INDEP(X,Y|Z)
holds. Condition (SCC) goes back to Reichenbach (1956).
(CE) (for "common effects"): The causes of a common effect are unconditionally inde-
pendent, but may be dependent conditional on a common causal successor. Thus in a [suf-
ficiently] complete CM of the form XZY, INDEP(X,Y) holds, and DEP(X,Y|Z) will
hold if the CM is faithful (see definition (3)). Condition (CE) was discovered by Berkson
(1946) and is called "Berkson's paradox".
The full content of these three principles is expressed as follows: "If DEP(X,Y|U), then
there is an X-Y-connecting path π such that (P)", where (P) is one of the clauses (SIC),
(SCC) or (CE) in (1), respectively.
3
The modularization of axiom (C) into three entailed
principles is important for our purpose of revision because it allows to alter one part of (C)
without changing the others.
A causal graph (model) is acyclic iff it contains no cyclic path XX. In this paper
we concentrate on acyclic graphs. For acyclic CMs, condition (C) is equivalent to the caus-
al Markov condition (M) (Pearl 2009, 16; SGS 2000, theorem 3.3):
(2) Definition: (V,E,P) satisfies the causal Markov condition (M) iff every XV is in-
dependent of any subset U V{X} of non-successors of X, conditional on X's parents:
3
Axiom (C) is stronger than the conjunction of these three principles, because in (C) the
clauses (SIC), (SCC) and (CE) are in the scope of the same existential quantifier over
X-Y-connecting paths π.
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INDEP(X,U|Par(X)).
4
The equivalence between (M) and (C) has been demonstrated by Verma and Pearl (s. Pearl
1988, 119f, th. 9) and Lauritzen et al. (1990, 50). The latter authors call (C) the global and
(M) the local Markov condition, since in its contraposed form, (C) asserts a (conditional)
independence for all d-separation relations of a causal graph, while (M) asserts such an
independence only for the d-separation relations between a variable X and its non-effects
conditional on its parents; the other independencies follow from these as probabilistic con-
sequences. For this reason, (C) expresses the full content of the causal Markov condition in
a more transparent way than (M). Moreover, the three principles (SIC), (SCC) and (CE)
are more directly contained in axiom (C) than in condition (M).
The explicit distinction between the definition of (C) and the axiom (C) in (1) impels us
to critically reflect on the problem of generality: Do all probabilistic dependencies between
analytically independent variables in fact result from causal d-connections? This has been a
question of debate for several decades. Two remarks on this debate are important for our
purpose:
First: Although the criticism was often directed against the Markov condition "in gen-
eral", our reconstruction of axiom (C) as entailing three independent conditions helps us to
locate the problem: What was under attack was not condition (SIC) or (CE), but only
(SCC) screening off by common causes.
Second: Defenders of TCN usually acknowledge that (SCC) fails in application to far-
distance (EPR/Bell) correlations in quantum mechanics, but regard (SCC) as valid for all
other domains (SGS 2000, 37f; Pearl 2009, 62). Only a few authors, foremost Cartwright
(e.g., 2007), have insisted that the causal Markov condition may fail for macrophysical
systems as well. In section 2 we will support this position, but since the counterexamples
only concern subcondition (SCC) of (C), we argue in sec. 3 that the Markov condition
should not be abandoned, but should be revised in certain ways.
Axiom (C) asserts an implication from probabilistic dependence to d-connection. The
converse implication from d-connection to probabilistic dependence is called the condition
of faithfulness (SGS 2000, 31, Zhang and Spirtes 2008, 24):
(3) Definition: A CM (V,E,P) satisfies the faithfulness condition (F) iff for all X, Y V
and U V{X,Y}: If X and Y are d-connected given U, then DEP(X,Y|U).
To avoid confusion, note that SGS (ibid.) define "faithfulness" as the conjunction of condi-
tions (C) and (F). We prefer definition (3) (which is also used in Zhang and Spirtes 2008,
247), because it logically separates (F) from (C). This is essential for our revision, which
does not concern (F) but only one part of (C) (i.e., (SCC)).
4
Condition (M) is explicated in terms of dependence between sets of variables. The lat-
ter notion reduces to dependence between variables as follows: DEP(U,V|W) iff
DEP(X,Y|W,U{X},V{Y}) for some XU and YV. Similarly, the notion of d-
connection can be extended to sets of variables.
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The probability distributions over the variables Xi of a CM conditional on their parents,
P(Xi|Par(Xi)), are called the parameters of the CM (Pearl 2009, 44). It is well known that
the faithfulness condition may fail for CMs whose parameters are fine-tuned in a such way
that the influences of two (or more) d-connecting paths are opposite and cancel each other
out. An example is presented in fig. 1.
Z
+
X
+
Y
Fig. 1: Unfaithful causal graph: DEP(X,Y|Z) but INDEP(X,Y), because the positive influ-
ence X+Y and the negative influence X+ZY cancel to zero.
5
Therefore condition (F) is not suited as a general axiom of TCN. Proponents of TCN
argue that the satisfaction of (F) is at least highly probable, because unfaithful CMs are
parameter instable, in the sense that their unfaithful independencies can be destroyed by
arbitrary small changes of the parameters of the CM (ibid., def. 2.4.1). This fact implies
that unfaithful independencies are statistically rare in all causal systems which are subject
to external noise
variables that are mutually independent and, thus, produce independent
variations of the system's parameters.
However, there is a weakening of (F) which holds in full generality and constitutes
TCN's second core axiom the condition of minimality (SGS 2000, sec. 3.4.2):
(4) Definition: A (C)-satisfying causal model (V,E,P) satisfies the minimality condition
(Min) iff no arrow can be omitted from E without violating condition (C).
Axiom: Every physically possible (C)-satisfying causal system satisfies (Min).
In contrast to axiom (C), axiom (Min) can justified by a methodological requirement that is
a variant of Ockham's razor: Causal arrows are only assumed if they are responsible for at
least some (conditional) probabilistic dependence, for values that are either factually real-
ized or can be brought about by interventions.
6
Schurz and Gebharter (2016, sec. 3) investigate the question whether the theory TCN
5
Here X +/ Y means that the arrow XY is responsible for a positive/negative de-
pendence of Y on X (as explained), conditional on par(Y){X} (cf. Schurz and Gebhar-
ter 2016, (9), theorem 2).
6
To ensure the compatibility of (Min) with the interventionist explication of a "direct
cause" (cf. Zhang and Spirtes 2011) we assume that values that can be brought by in-
tervention have a positive probability.
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has empirical content by which it can be independently tested. Since causal arrows are
regarded as theoretical (unobservable) relations, the TCN would have empirical content if
it excluded logically possible probability distributions, i.e., if there were probability mod-
els (V,P) that cannot be extended to TCN-satisfying causal models. The authors prove that
the core of TCN, axiomatized by (C)+(Min), is without empirical content. If an axiom of
cautious faithfulness is added to (C)+(Min) "cautious" in the sense that it asserts faithful-
ness merely with high probability this extended version of TCN acquires weak probabil-
istic content. More empirical content is obtained if one adds to TCN the axiom of temporal
forward-directedness (T). This axiom applies to CMs whose variables are event-variables,
i.e., variables Xi whose values are possible events having a location in time, t(Xi). For
these CMs condition (T) states that (for all X, Y V) XY implies t(X) < t(Y), i.e., caus-
es precede their effects in time. The extension of (C)+(F) by (T) excludes a variety of logi-
cally possible probability distributions (ibid., sec. 3.3).
Several objections against the truth of the causal Markov condition have been success-
fully defeated by proponents of TCN (cf. SGS 2000, 59-63; Pearl 2009, 62; Hitchcock
2010). In particular, it has been shown that (C) can only be true if
(a) the variables in V are analytically independent; otherwise one obtains "arrows" on
purely conceptual grounds (Hitchcock 2010, §2.3, §2.10), and
(b) the description of the causal system is sufficiently refined (cf. Haussman and
Woodward 1999, 527-9).
Since the omission of certain variables or edges from a (C)-satisfying causal model can
lead to a violation of (C), axiom (C) refers prima facie to physically possible CMs that are
complete in the following sense:
(5) Definition: (V,E,P) is a complete physically possible CM iff (V,E,P) represents a causal
system in a physically possible world w and every variable in w that is a direct cause or
effect of a variable in V is itself in V.
Unfortunately, complete causal systems are typically not epistemically accessible. What
does condition (C) entail for coarse-grained descriptions of causal systems that omit some
detail and, thus, are incomplete?
A structural coarsening of a CM removes some variables and their edges, but preserves
cause-effect relations between the remaining variables. It is well known that the satisfac-
tion of condition (C) is not preserved by every structural coarsening of a CM, but only by
structural coarsenings that are causally sufficient, i.e., do not remove a common cause
(SGS 2000, 22, 27). However, there is a second kind of coarsening which plays a role in
the next section and is called conceptual coarsening. Here a CM is described with the help
of a compound variable Z that is definable in terms of more fine-grained variables
X1,,Xn; we write Z = f(X1,,Xn). When a model is conceptually coarsened by a com-
pound variable Z, then the set of defining variables X1,,Xn is replaced in V by the com-
pound variable Z and an arrow between a variable Y and Z is added whenever the original
model contained a corresponding arrow between Y and some defining variable of Z. This
operation is illustrated in fig. 2.
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G1: Y1 X1 X2 X3 Y2 G2: Y1 Z Y2
Fig. 2: Coarsening of causal graph G1 by the compound variable Z = f(X1,X2,X3).
Assume that in the refined model M1 of fig. 2 axiom (C) is violated, because
DEP(X1,X3|X2) holds (since X2 is an interactive cause; see sec. 2). Nevertheless IN-
DEP(Y1,Y2|X1,X2,X3) holds in M1, thus by assuming f to be injective, INDEP(Y1,Y2|Z)
will hold in M2 (since each value triple (x1,x2,x3) is uniquely translated into one Z-value z
= f(x1,x2,x3)). So M2 satisfies (C), although its conceptual refinement M1 fails to satisfy
(C). This shows that a conceptual coarsening of a CM may hide a violation of (C).
To sum up, neither does the satisfaction of (C) by a complete causal model M* imply
that a coarsening of it will satisfy (C), nor does the violation of (C) by M* imply that a
coarsening of M* will violate (C). What can be guaranteed is that a CM is "sufficiently
complete", in the sense of not distorting the true laws of causality, if its variables describe
reality at the most fine-grained level and do not omit true common causes. Turning this
into a definition would not be very useful, however, since most descriptions of reality are
coarse-grained. All that can be said of coarse-grained descriptions is this:
(6) If (C) is satisfied by the complete causal model system CM* of a world w, then every
coarse-grained CM that is true in w has a true refinement in w that satisfies (C).
Although condition (6) is analytically true, it gives us a heuristics for testing (C):
(7) Test heuristics for (C): If a true causal model M is found such that not only M but eve-
ry true refinement of M that is consistent with the established background knowledge vio-
lates (C), then the truth of (C) in the actual world is strongly disconfirmed. On the other
hand, if [almost] all causal models considered so far satisfy (C) for all plausible refine-
ments of them, then (C) is confirmed.
In the next section we will make use of this heuristics.
2. Interactive Causes: Violation of (SCC) in Indeterministic Decay Processes
Given that axiom (C) is empirically empty in isolation, how can the truth of (C) be tested
by empirical means at all? Following from the results concerning the empirical content of
TCN mentioned in sec. 1, there are two ways of testing axiom (C): either one strengthens
(C) by condition (F) or by condition (T). It is natural to assume condition (T) in the coun-
terexample to which we now turn, namely the example of spontaneous decay. However,
our arguments do not presuppose this condition: we shall see that even without assuming
(T) certain causal structures involving spontaneous decay violate axiom (C) with high
plausibility, because they cannot satisfy (C) in a faithful way.
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The example of spontaneous decay has been given in the literature several times, but it
was often presented as if it were a specific aspect of quantum mechanics (cf. van Fraassen
1980, 29f; Arntzenius 2010, sec. 2.1). In contrast, Cartwright (2007, 122) has emphasized
that the counterexample only requires an indeterministic decay process. Indeterministic
processes are not only found in quantum physics, but also in classical physics.
7
Consider a particle (of type) q that decays with a certain probability p into two particles
(of types) q1 and q2. Since the decay is objectively indetermined, i.e., the probability of q's
decay conditional on all common causes in the past is (significantly) smaller than 1; how-
ever, if the particle decays, then it always (or almost always) decays into q1 and q2. Or at
least: If q decays and one decay product is q1, then the other decay product is (almost) al-
ways q2. Examples are radioactive decay (e.g., the -decay of uranium U238 Th234 +
He4), spontaneous chemical decay (e.g., the decay of ethene C2H4 CH4 + C), or in eve-
ryday life the spontaneous breaking of an eroded mountain rock into two pieces, etc.
What leads to a violation of (SCC) is the combination of two factors, (i) indeterminism
plus (ii) the law describing a decay. Physically speaking the latter is an instance of a con-
servation law. Generally speaking, a conservation law for a quantity X says that the sum of
the X-values of all parts of a closed system remains constant through time.
Classical mac-
rophysical or chemical decay is described by the conservation of mass (m): if q decays into
q1 and q2, then m(q) = m(q1)+m(q2). This explains qualitative decay laws as follows: If q
decays into two parts and one part is q1, then the other one must be q2, because q2 is the
only physically possible decay component for that type of decay that conserves mass, via
m(q2) = m(q) m(q1). More general conservation laws of physics, such as the conservation
of energy, momentum, and charge, give rise to a more general notion of indeterministic
decay that is introduced at the end of this section. In the next couple of paragraphs we dis-
cuss simple qualitative decay.
To reconstruct qualitative decay processes we assume binary variables Qi that apply to
spatiotemporal locations of a small but finite size as their domain. The value of Qi()
indicates whether a given type of particle qi is present (Qi() = qi) or absent (Qi()= off) at
the location or in a certain neighborhood f() of that is determined by a spatiotempo-
ral function f; in the latter case the variable carries "f" as a second index: Qf,i. In the follo-
wing we consider a symmetric decay process in which a particle q decays into two type-
identical particles ql and qr that are moving away from each other in opposite directions
(see fig. 3). We let Ql() be the binary variable denoting the (non-)occurrence of a particle
ql shortly after the decay-time at the left spatial side of and Qr() the (non-)occurrence of
qr at the right side of .
7
Since certain classical differential equations admit of trajectory branching (cf. Earman
1986, ch. III; Norton 2008).
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space
ql Ql
q Q
qr Qr
time
Fig. 3: Indeterministic decay: Spatiotemporal diagram (left) and causal graph (right). Par-
ticle q (described by variable Q) decays into decay-products ql and qr (described by Ql and
Qr, respectively). (SCC) fails, since neither Q nor any extended set of common causes in
the past screens off Ql from Qr. "Q" is called an interactive (common) cause and is graph-
ically represented by the arc.
Since neither the state of the particle Q nor any further variable before the decay
screens off Ql from Qr, condition (SCC) of (C) is violated: P(ql|qr,q) > P(ql|q), i.e.,
DEP(Ql,Qr|Q). Moreover, DEP(Q1,Q|Past) holds for any set "Past" of variables describing
events before the decay time.
Note that for the violation of (SSC) one need not assume that the conservation law
holds with probability 1; it suffices that it holds with a probability p' higher than the decay
probability p. We follow Salmon (1984, 158-77) and call a common cause that does not
screen off its effects an interactive (common) cause and a common cause that does screen
off its effects a conjunctive cause.
There are four prominent defense strategies for (SCC) against this type of counterex-
amples:
(1.) Value-refinement of the interactive cause: Salmon's example of an interactive
cause was one of deterministic physics. SGS (1993, 63) argued that if in that example the
state of particle q immediately before the decay were described with maximal precision, it
would imply the values of Ql and Qr with probability 1 and, thus, screen them off. Howev-
er, in our example particle q decays only with a certain probability, so this defense does not
apply.
(2.) Linking the effects: One way to reconcile interactive forks with condition (SCC) is
to add a direct causal link between the two effects (see fig. 4(a)). This possibility is typical-
ly excluded in the given examples of indeterministic decay because the two effects occur
simultaneously; so a causal link between them would violate condition (T), which holds in
classical and relativistic physics because of their assumption of locality. In contrast, in
quantum mechanics it is debated whether non-local causal links should be admitted or not
(cf. Wood and Spekkens 2012, 14ff, Näger 2016).
Even without the assumption of condition (T), the causal reconstruction in fig. 4(a)
leads to the problem of unexplained violations of faithfulness. There is a crucial difference
between the conditional dependence between effects of an interactive cause and that be-
tween effects of a common cause with an additional causal link between them. In a faithful
reconstruction of the latter model this causal link can be used to transmit the influence of
an intervention on one effect to the other effect conditional on the common cause, but this
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is impossible in the case of an interactive fork. In our example of decay such an interven-
tion could be realized by positioning a device at the left side of that emits particles of
type ql so that the probability of finding p1 conditional on presence of q is increased, com-
pared to the distribution without this device. Formally, PI(ql|q) > P(ql|q), where P is the pre-
intervention and PI the post-intervention distribution. While in the case of indeterministic
decay, the post-intervention probability of the other decay product qr would be unchanged
(PI(qr|q) = P(qr|q)), it would be increased in a faithful version of fig. 4(a) (PI(qr|q) >
P(qr|q)). In what follows we call this characteristic feature of interactive forks the non-
transmission condition. In section 3 we will demonstrate that a (C)-satisfying reconstruc-
tion of this feature is not possible without postulating mysterious unfaithful independen-
cies.
(3.) Introduction of a hidden common cause: Pearl (2000, 62, fig. 2.6) suggested to
explain DEP(Ql,Qr|Q) by assuming an additional hidden common cause Q* which together
with Q screens off Ql from Qr (INDEP(Ql,Qr|Q,Q*); see fig. 4(b)). However, this defense
strategy cannot explain a second characteristic feature: Interactive cause variables screen
off their common effects when their value is "off", i.e., when the decaying particle or con-
served quantity is absent. To see this, assume that even if there is no particle q in the spati-
otemporal region there is a small probability that a subparticle ql/r spontaneously ap-
pears in the left/right neighborhood of , because it has moved in from the environment.
Since the spontaneous appearances of ql and qr are mutually independent, IN-
DEP(Ql,Qr|Q=off) must hold. However, this feature is not explained by the graph in fig.
4(b): Since the two common causes Q and Q* are causally unconnected, Q=off cannot en-
tirely screen off Ql from Qr because of the influence of Q*. The common cause scenario
could only explain this feature if a directed arrow led from Q to Q* that inactivates Q*
whenever Q is absent, P(Q*=off|Q=off) = 1. But then Q* would play the role of an inter-
mediate cause, which is the fourth defense strategy to be discussed below. We call this
second characteristic feature of interactive causes the absence-condition. It becomes im-
portant in a revised search procedure for interactive CMs.
Ql Ql Ql
Q Q Q* Q Q*
Qr Qr Qr
(a) Linking the effects (b) Hidden common cause (c) Hidden intermediate cause
Fig. 4: Defense strategies against the counterexample in fig. 3.
(4.) Introduction of a hidden intermediate cause: Even if the particle decays with a
probability of less than one, many authors have argued for the existence of some interme-
diate cause q* (described by variable Q*) in the form of a breaking event or state of the
particle immediately before the decay, so that q* causes the decay with certainty and,
hence, Q* screens off Ql from Qr (see fig. 4(c)). This defense strategy is, for example, used
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by Haussman and Woodward (1999, 563) against Cartwright's counterexample (2007,
122), in which a chemical process C succeeds 80% of the time and when it succeeds it in-
variably produces two end products, a chemical X and a pollutant Y. These authors assume
a certain intermediate event, the "operation of firing F", that deterministically produces X
and Y and screens off X from Y.
In this defense strategy the idea is to try to find a description of the decay process that
makes it deterministic. The refutation of this strategy is more involved. We will present
three arguments to show that this strategy either violates the assumption of indeterminism
or rests on the artificial effects of a too coarse-grained description.
Argument 1: The first argument says that a breaking event is nothing but a temporal
succession of two consecutive events or states: At time t, the particle q is still in the unbro-
ken state, while at the next time point, it is in the "broken state", which means that instead
of q we now have the two decay products ql and qr, possibly very close together, but al-
ready separated. If we reconstruct this in a discrete time, the picture is as in fig. 5:
"breaking event b"
qr qr qr
q q q ql ql ql
time
t t+1
last time t when first time when q is
q is unbroken replaced by ql, qr.
Fig. 5: Reconstruction of a "breaking event" in discrete time.
Since the breaking event "b(t,t+1)" is a compound of both the cause and its direct effects, it
screens off its direct effects trivially, i.e., INDEP(Ql(t+1),Qr(t+1)|b(t,t+1)) holds for analyt-
ical reasons; the screening off of the indirect effects follows from this by condition (SIC).
As soon as we refine the description and discriminate between the events at time t and that
at time t+1, the screening off disappears.
Proponents of this defense strategy tend to reply to this argument with the objection
that time is continuous; so there are no such two consecutive time points t and t+1. A clos-
er analysis shows that the continuity assumption cannot rectify the violation of the causal
Markov condition. If time is continuous, two cases are possible:
Either there is a last time point tR at which q is unbroken, while for all times t' > t the
particle is broken. Thus the set of times for which the particle is unbroken forms a right-
closed interval and the set of times for which it is broken an adjacent left-open interval; see
fig. 6(a). Then for all times t '> t Q(t) does not screen off Ql(t') from Qr(t'), which is a viola-
tion of condition (SSC).
Or alternatively there is a first time point t|R at which q is replaced by its decay-
products ql and qr. Then the set of times for which the particle is unbroken forms a right-
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open interval and the set of times for which it is broken an adjacent left-closed interval; see
fig. 6(b). Then for all times t '< t Q(t') does not screen off Ql(t) from Qr(t), which is again a
violation of (SSC).
qr qr qr qr qr qr
q q q ]( ql ql ql q q q )[ ql ql ql
t time t time
(a) t = last time at which q is unbroken. (b) t = first time at which q is broken.
Unbroken state: right-closed interval "]" Unbroken state: right-open interval ")"
Broken state: left-open interval "(" Broken state: left-closed interval "["
Fig. 6: Two possible reconstructions of a "breaking event" in continuous time.
Argument 2: The first argument shows that for every analysis of a breaking event as a
succession of fine-grained events condition (SSC) fails. Some proponents of (SCC) reply
to this argument as follows: Although this may be so, it is at least possible to describe the
causal structure in such a way that (SSC) holds, namely by using the compound variable of
a "breaking event". At this point our test heuristics of sec. 1, condition (7), comes into play.
What the defender of (SCC) says with this objection is, in effect, that (SCC) holds merely
because the description is coarse-grained and hides the failure of (SCC), which appears in
all refined models that decompose the breaking event into a succession of fine-grained
events. By test condition (7) this is strong reason to suppose that condition (SCC) is violat-
ed in the actual world.
Argument 3: The objection remains that the "breaking event" is more than an instanta-
neous event consisting of a mere succession of an unbroken and a broken state. Rather it
refers to a specific pre-breaking stage q* which the particle enters shortly before it decays
see fig. 7.
ql
q q* qr
time
Fig. 7: The particle q enters a pre-breaking state q* before it decays.
Even if this assumption is made, the analysis given in the previous argument applies: Ei-
ther there is a first time point t at which particle is broken, i.e., q* is replaced by q1 and q2,
or there is a last time point at which the particle is unbroken, i.e., q* exists. If the decay is
indeterministic then in both cases condition (SCC) is violated. The only way to save condi-
tion (SCC) is to assume (as in argument 1) that the transition from the pre-breaking state
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q* to the decay products ql and qr is deterministic. But this is implausible, since according
to contemporary scientific understanding, radioactive decays or the breaking of chemical
bonds (etc.) are indeterministic processes. Moreover, why should we assume that although
the beginning (the left end) of the particle's pre-breaking stage is an indeterministic process
(P(q*|q) < 1), its ending (the right end) should be a deterministic process? This seems too
strange to be plausible: If the decay process is indeterministic, then it is indeterministic "on
both sides".
To summarize, in all plausible reconstructions of indeterministic decay processes, the
particle's state before the decay does not screen off the states of its decay products.
So far we have only analyzed simple (qualitative) indeterministic decay, in which the
decay itself occurs spontaneously. However, the same reasoning applies in cases of gener-
alized indeterministic decay: Here the decay event may have a deterministic cause, but a
conserved quantity X of the unbroken particle q is indeterministically distributed over its
decay products, in the sense that (i) P(X(qr)=x2 | X(q)=x1) is small but (ii) P(X(qr)=x2 |
X(q)=x1, X(qr) = x1x2) is (close to) one. (i) and (ii) imply the interactive fork condition
DEP(X(qr)|X(q),X(ql)). As an example consider the generalized decay of a high energy
gamma particle (into a positron (e+) and an electron (e-), which is also called "pair crea-
tion". It is caused by the clash of a particle against an atomic nucleus (a). Since the mo-
mentum ( p
) of the particle is absorbed by a ( p (a) = p ()), the total momentum of p
and e must be zero ( p
(e+)+ p
(e-) = 0). This means that e+ and e- fly apart in opposite di-
rections; but the particular direction (and momentum) is undetermined. This leads to the
interactive fork equations (i) and (ii) above, if we substitute for q, e+ for ql, e- for qr and
p
for X.
The notion of generalized indeterministic decay also applies to far-distance correlations
in quantum-mechanical EPR/Bell experiments. Here a source emits a pair of entangled
particles (e.g., electrons) flying in opposite directions. The interactive cause is the state of
the entangled two-electron system q, whose total spin in the direction of the z-axis is con-
served. Upon measurement at one of the two "sides" of the spatially extended two-electron
system, q's wave function collapses into a statistical ensemble of the states of the two sepa-
rate electrons ql and qr. This collapse is described as a "generalized decay". While condi-
tional on the measurement this decay obtains with certainty, the total spin is indeterministi-
cally distributed over the two electrons, satisfying the conservation law S(ql)+S(qr) = S(q).
Depending on the states (polarization angles) of the measurement devices Ml and Mr,
measurement of S(ql) and S(qr) yields the results R1 and Rr, respectively (with Val(Ri) =
{"up","down"}), whose distribution satisfies Bell's famous inequality-violating correla-
tions: DEP(R(ql),R(qr)|M1,M2 ,S(q)) (cf. Wood and Spekkens 2012). In sec. 3 we will argue
that this correlation can be faithfully explained by an interactive cause, but not by a com-
mon cause model. The difference from classical indeterministic decay does not concern the
basic structure of an interactive fork, but rather the non-local nature of the entangled sys-
tem whose wave function stretches over a huge distance, although it is instantaneously
influenced by measurements on either side of the electron-pair.
The conclusion of this section is that in indeterministic decay processes (in the simple
or generalized sense) condition (SCC) is violated. Philosophers who accept this conclusion
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may nevertheless defend the causal Markov condition by pointing out that indeterministic
decays are exceptions that do not arise in everyday life or in the special (life or social) sci-
ences. This does not seem to be true, either. An everyday life example of a violation of
(SCC) is a rock q in the Alps from which a piece q1 breaks off spontaneously, leaving the
now smaller rock q2 behind, with mass(q) = mass(q1) + mass(q2). Given rock q at time t,
the probability that in the next moment of time t+1 a piece q1 breaks off is low; but given
q2 at time t+1, with near certainty the missing piece q1 is in the (dangerous) state of falling
downwards at time t+1.
Concerning the social sciences, we conjecture that quantities possessed by social
groups (national wealth, etc.) may undergo processes of "generalized decay": often the
total amount of such a quantity remains fairly constant, while its social distribution under-
goes unpredictable changes. An elaboration of this conjecture is left to the future.
In conclusion, it appears that violations of condition (SCC) due to interactive causes
may occur in all domains. In the next section we develop a revision of condition (C) that
accounts for interactive causes.
3. Revision of TCN
The minimal revision of TCN that accounts for interactive causes is to exempt interactive
causes from the d-connection condition (C), or more precisely, to exempt them from the set
of nodes whose presence in the conditioning set blocks a d-connection. In what follows we
call the minimal revisions of the notions of d-connection, condition (C) and faithfulness
"d*-connection", "condition (C*)" and "d*-faithfulness", respectively. The unstarred and
starred notions coincide for CMs without interactive causes, but differ in regard to interac-
tive causes. We will argue that the minimal revision is the right revision, except for an ad-
ditional constraint that accounts for the absence condition (recall sec. 2).
We display an interactive cause graphically by an arc that connects the arrows pointing
towards its direct effects.
8
We call such an arc an interactive arc and assume that effects
connected by such an arc are d*-connected conditional on their interactive cause (see fig.
8(a)); we also say they are "interactively d*-connected". If the interactive causal model is
d*-faithful, then two interactively d*-connected effects will be probabilistically dependent
conditional on their interactive cause; we say in this case that they are "interactively (prob-
abilistically) dependent". Moreover, an interactive cause may have more than two effects,
as in fig 8(b): an example is the -decay of a neutron into proton, electron and anti-
neutrino.
8
Arcs are more specific than double-headed arrows, which are used in the literature to
represent unexplained correlations of different sorts, e.g., due to hidden conjunctive
causes. See Richardson (2003).
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Y1 Y2 Y1 Y2 Y3
X X
(a) (b)
Fig. 8: Interactive forks are displayed by arcs connecting interacting arrows.
In sec. 2 we characterized the correlations between interactively d*-connected effects
by the non-transmission condition, which distinguishes them from the conditional correla-
tions between effects of a common cause that result from of a causal link between them. In
the latter case, the conditional dependence between the effects can be used to propagate the
influence of an intervention on the causing effect to the other effect (provided the causal
model is faithful). This is depicted in fig. 9(a), where the variable Z1 figures as an interven-
tion variable on the causing effect Y1: the result is DEP(Y2,Z1|X). In contrast, the condi-
tional correlation between the effects of interactive causes cannot be used to propagate the
influence of an intervention: in fig. 9(b) INDEP(Y2,Z1|X) holds.
Y1 Y2 Y1 Y2
Z1 X Z1 X
(a) (b)
Fig. 9: Distinguishing correlations produced by directed arrows (a) from correlations
produced by interactive causes (b). In both CMs (which are assumed to be faithful) we
have DEP(Y2,Y1|X), but only in (a) do we have DEP(Y2,Z1|X), i.e., Z1 can be used as in-
tervention variable on Y2 w.r.t. Y1. In (b) INDEP(Y2,Z1|X) holds (non-transmission condi-
tion).
Condition (C*) (the minimal revision of (C)) is sufficient to account for the behavior
displayed in fig. 9. It implies for fig. 9(a) that Y2 is d*-connected (as well as d-connected)
with Z1 given and given X, along the path Z1Y1Y2. For fig. 9(b) it yields that Y2 is
not d*-connected (nor d-connected) with Z1 given or given X. To ensure d*-connection,
the common effect Y1 has to be included in the conditioning set, i.e., in fig. 9(b) Y2 is d*-
connected (and d-connected) with Z1 given Y1.
Condition (C*) has the further consequence that in fig. 9(b) Y2 is d*-connected with Z1
given {X,Y1}, which is not the case for d-connection in fig. 9(b) nor in fig. 9(a). Do we
have a corresponding probabilistic dependence DEP(Y2,Z1|X,Y1) in fig. 9(b)? If the CM is
d*-faithful, the answer is "yes". Here is an intuitive example that illustrates this depend-
ence: Assume that the interactive cause Y1XY2 in fig. 9(b) describes the spontaneous
decay of particle q into q1 and qr, but the conservation law is not entirely deterministic
(P(ql|qr,q) < 1) and Z1 is an additional factor whose presence increases the probability of
ql's presence (Y1=ql) after the decay. "Z=on" may represent a field that attracts ql particles
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and increases the probability that they fly into region from outside it. Assume the parti-
cle decays (X=q) and the right decay product is absent (Y2=off). Then the absence of qr in
spite of the presence of q increases the probability that the presence of ql was caused by
Z1=on. This means that P(Z1=on | Y2=off, X=q, Y1=ql) > P(Z1=on | X=q, Y1=ql), i.e.,
DEP(Z1,Y2|X,Y1). Thus the revised condition (C*) yields the correct result.
A non-interactive and d-faithful reconstruction of the (in)dependences characterizing
fig. 9(b) would be possible by inverting the direction of the causal arrow Y1Y2 in the
CM of fig. 9(a); the resulting structure would explain DEP(Y1,Y2|X), INDEP(Z1,Y2|X) and
DEP(Z1,Y2|X,Y1). This is no longer possible, however, in the CM in fig. 10(a), which ex-
tends the CM in fig. 9(b) by adding a second cause Z2 to Y2: Z2Y2. The (in)dependences
characterizing this CM cannot be described by a non-interactive CM in a faithful way. To
explain DEP(Y1,Y2|X), INDEP(Z1,Y2|X) and DEP(Z1,Y2|X,Y1) in a d-faithful way, we
would have to add the arrow Y2Y1 to the CM without the arc above X. But this would
turn INDEP(Z2,Y1|X) into an d-unfaithful independence, because then Z2 is d-connected
with Y1 given X. Moreover, we would additionally need the arrow Z2Y1 to explain
DEP(Z2,Y1|X,Y2). This unfaithful CM is depicted in fig. 10(b); the bold arrows generate
unfaithfulness.
Y1 Y2 Y1 Y2
Z1 X Z2 Z1 X Z2
(a) (b)
Fig. 10: (a) A d*-faithful interactive CM whose (in)dependencies have no faithful non-
interactive modeling.(b) A d-unfaithful CM of the (in)dependencies characterizing (a)
(bold arrows generate unfaithfulness).
The example of fig. 10(a) shows that there are d*-faithful interactive CMs that admit of
no faithful non-interactive reconstruction. Thus, misidentifying an interactive as a conjunc-
tive cause may lead into unfaithfulness. This fact may explain recent findings of Wood and
Spekkens (2012, 2) and Näger (2016) who applied TCN to quantum-mechanical EPR/Bell
correlations and conclude, in the words of Näger (ibid., 12), that "the causal faithfulness
condition and the causal Markov condition cannot both hold in the quantum realm." Recall
the EPR/Bell experiment described in sec. 2: If we identify the variable S(q) of this exper-
iment (describing the spin state of the entangled electron-pair q before measurement) with
the variable X in fig. 10(a), identify the variables M1 and M2 describing the state of the
measurement devices with Z1 and Z2 in fig. 10(a), and the variables R(q1) and R(q2) de-
scribing the measurement results with Y1 and Y2 in fig. 10(a), then we obtain the causal
structure describing the EPR/Bell experiment. The non-transmission condition that charac-
terizes interactive causes, INDEP(Z1,Y2|X), corresponds to the famous "no signaling" con-
dition. As we explained at hand of fig. 10(b), a (C)-satisfying reconstruction of the proba-
bilistic (in)dependences in this experiment without assuming a hidden common cause is
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only possible if one introduces the additional arrows Y2Y1 and Z2Y1 that imply "su-
perluminal causation" and generate unfaithfulness (Wood and Spekkens 2012, fig. 23).
9
In
contrast, the reconstruction of the EPR/Bell experiment as an interactive CM can explain
all (in)dependencies of this experiment in a d*-faithful way. These remarks are intended as
promissory notes concerning the applicability of the proposed account to quantum mechan-
ics; working out the details would require a paper of its own.
So far we have considered interactive effects having a further cause. In fig. 11 we con-
sider two interactive forks that are connected via a common effect. Here Z1 and Z2 are not
d*-connected given {X1,X2}, but are d*-connected given {X1,X2,Y}, i.e., conditional on
their common effect. By (C*) we get INDEP(Z1,Z2|X1,X2) and by d*-faithfulness
DEP(Z1,Z2|X1,X2,Y). As an example, assume that two particles x1 and x2 decay into the
partially overlapping decay products z1, y and y, z2, respectively. The presence of z2 condi-
tional on x2 makes y's presence more probable; but this probability increase is not propa-
gated to z1 conditional on x1, because the decay of x1 is independent from that of x2. How-
ever, conditional on the presence of y, x1 and x2, the absence of z2 increases the probability
that z1 is present because x1 decayed. Again the minimal revision gives the right result.
Z1 Y Z2
X1 X2
Fig. 11: Two interactive causes may share an effect: Z1 and Z2 are d*-connected and de-
pendent only if Y is in the conditioning set.
We finally turn to interactive causes with more than two effects, as in fig. 8(b) above.
We require that if more than two effects are connected by one arc, then each effect be in-
teractively dependent on at least one other effect. We consider this as the extension of the
minimality axiom for d*-connections: Each d*-connection by an interactive arc must pro-
duce at least one interactive probabilistic dependence, so that removing this arc causes a
violation of condition (C*). On the other hand, interactive correlations within one arc need
not always be transitive, so that all pairs of effects within one arc become interactively
correlated; this is only the case if the interactive CM is d*-faithful.
We admit the possibility that the same interactive cause produces two (or more) sets of
interactive effects in separate interactive events. This is displayed in fig. 12, in which
{X1,X2} and {X3,X4} are sets of interactively d*-connected effects of Z; moreover, X5 is
an additional conjunctive effect of Z. As an example, consider high-energy - rays (X) that
create two different kinds of particle-pairs, electron-positrons (X1,X2) and muon-antimuons
(X3,X4) and in addition produce heat (X5). Following from the above requirement on arcs,
9
Even the assumption of a hidden common cause of Y1 and Y2 cannot explain the distri-
bution over {Z1,Y1,X,Y2,Z2}, because it violates Bell's inequalities, but it opens two
further possibilities of unfaithful (C)-satisfying reconstructions, "superdeterminism"
and "retrocausation" (Wood and Spekkens 2012, 14, fig.s 25-27).
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different sets of interactive effects of one interactive cause are disjoint, i.e., a variable in
one set is merely conjunctively correlated with the variables in another set.
X1 X2 X3 X4 X5
Z
Fig. 12: One interactive cause causes two interactive pairs of effects and one ordinary
effect. Variables not connected by an interactive arc are conjunctively related.
We have seen that the proposed minimal revision of TCN handles the discussed exam-
ples in a satisfactory way. Now we are ready to state the exact formulation of our revision
of TCN. Recall that the absence of the interactive cause is expressed by letting the interac-
tive cause variable X take the value "off", X() = off, meaning that the conserved quantity
represented by X is not present in the region . If the conserved quantity is positive (un-
signed) as in the case of mass, energy or momentum may the value "off" be identified
with the value "zero". In application to signed magnitudes such as charge this identification
fails, since a particle may have a total charge of zero, but consist of two subparticles hav-
ing a positive and negative charge of equal amount; in this case the conserved quantity is
present, although its total value is zero.
Based on the preceding considerations we formally define the notion of an interactive
CMs as follows ("Pow" for "power set"):
(7) Definition: An interactive causal model, for short an ICM, is a quadruple (V,E,I,P)
such that:
(7.1) (V,E,P) is an ordinary causal model (as defined in sec. 1).
(7.2) I VPow(V) is a nonempty set of pairs of the form (X,U), so-called interactive
forks, satisfying:
(a) XV and off Val(X) (possession of an absence-value),
(b) for all YU XY E, and
(c) for all (X,Ui), (X,Uj) I: UiUj = (disjointness of arcs).
The node X of (X,U) is (called) an interactive cause, the set U an interactive arc, and the
interactive fork (X,U) is graphically displayed as follows:
Y1
X where U = {Y1,,Yn}
Yn
(7.3) (V,E,I) is called the interactive causal graph (ICG) of the ICM.
For U V a subset of variables of an ICG (V,E,I), I(U) U denotes U's subset of in-
teractive causes in (V,E,I). The notion I(U)=off abbreviates the conjunction of the condi-
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tions X=off for all XI(U) (should I(U) be empty, the condition I(U)=off becomes vacu-
ous). An interactive cause Z lies on a path iff carries YZX, i.e., Z's outgoing ar-
rows on this path are connected by an arc (which may connect other effects), or formally, if
for some U V, (Z,U) I with {X,Y} U. A conjunctive cause Z lies on path iff
carries YZX and Z carries no arc, i.e., there exists no U V with (Z,U) I and
{X,Y} U. The other graph-theoretic notions remain unchanged.
Our notion of d*-connection exempts interactive causes from clause (a):
(8) Definition: Two nodes X, Y of an interactive graph (V,E,I) are d*-connected given a
set of nodes U iff they are connected by a path in (V,E) such that
(a) no intermediate cause or conjunctive common cause on is in U, and
(b) every common effect on is in U or has an effect in U.
The next definition explicates the revised causal d*-connection condition. It consists of the
minimal revision (i) together with the absence condition (ii):
(9) Definition: An interactive causal model (V,E,I,P) satisfies the causal d*-connection
condition (C*) iff for all X,YV and U V{X,Y}:
(i) if DEP(X,Y|U), then X and Y are d*-connected given U, and
(ii) if DEP(X,Y|U, I(U)=off), then X and Y are d-connected given U.
Axiom: Every physically possible system (V,E,P) that is [sufficiently] complete satisfies
condition (C*) .
Observe that d-connection implies d*-connection; so if a CM satisfies (C), it satisfies
(C*).
Condition (9)(ii) is equivalent to "if X and Y are d-separated given U, then IN-
DEP(X,Y|U,I(U)=off)". This condition breaks the symmetry between (C) and (C*), but it is
useful for the purpose of constructing a discovery algorithm for ICMs.
(C*) is the revised global Markov condition. For finite acyclic ICMs (C*) is equivalent
to a correspondingly revised local Markov condition (M*). We say that Y is an interactive
non-successor of X if Y is neither an ordinary X-successor nor a successor of a variable Z
that is connected with X by an interactive arc:
(10) Definition: An interactive causal model (V,E,I,P) satisfies the revised Markov condi-
tion (M*) iff every XV is
(i) independent of any subset of interactive non-successors of X conditional on Par(X), and
(ii) independent of any subset of (ordinary) non-successors of X conditional on Par(X) and
the condition I(Par(X))=off.
Theorem 1: For acyclic finite ICMs, conditions (C*) and (M*) are equivalent. (Proof
omitted)
The conditions of d*-faithfulness and d*-minimality are defined in the obvious way:
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(11) Definition: (11.1) An ICM (V,E,I,P) satisfies the condition of d*-faithfulness (F*) iff
for all X,YV and U V{X,Y}: If X and Y are d*-connected given U, then
DEP(X,Y|U).
(11.2) A (C*)-satisfying ICM (V,E,I,P) satisfies the condition of d*-minimality (Min*) iff
no arrow can be removed from E and no interactive fork from I without violating condition
(C*).
Axiom: Every physically possible (C*)-satisfying causal system satisfies (Min*).
Axioms (C*) and (Min*) form the core of TCN*. Possible extensions are obtained by add-
ing a cautious version of (F*) or condition (T).
4. Conclusion and Outlook: The Discovery Problem for TCN*
In this paper we argued that processes of indeterministic decay involve interactive causes.
They violate the principle of "screening-off by common causes", which is entailed by
causal Markov condition, the core axiom of the theory of causal nets (TCN). We showed
that there are interactive CMs that admit of no faithful non-interactive reconstruction. We
developed a revised version the TCN, TCN*, that correctly predicts the probabilistic
(in)dependencies of interactive CMs and coincides with TCN for CMs without interactive
causes.
We cannot discuss the discovery problem for TCN*. Instead, we confine ourselves to
brief observations. Recall the SGS discovery algorithm for non-interactive CMs (SGS
1993, 114; Pearl 2000, 50): It first finds the unique undirected causal graph (assuming
conditions (C) and (F)) and then orientates some edges as common effects. We suggest that
a revised discovery algorithm for ICMs can be obtained from the SGS algorithm by adding
a third step, roughly described as follows. For every undirected triangle (X,Y,Z) in the un-
directed graph and node X in this triangle, as in fig. 13(a), the algorithm checks whether an
interactive cause interpretation of X is possible, as in fig. 13(b). This presupposes that (i)
the absence condition holds (offVal(X) and INDEP(Y,Z|X=off,U) for some UV) and (ii)
edges between X, Y and neighboring nodes that have already been oriented are coherent
with the non-transmission condition.
Y Z Y Z
X X
(a) (b)
Fig. 13: Additional discovery step for ICMs: Can an undirected triangle (a) be recon-
structed as an interactive fork (b)?
The design of a revised discovery algorithm along these lines is a task for the future.
Copyright Philosophy of Science 2016
Preprint (not copyedited or formatted)
Please use DOI when citing or quoting
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